Determinant#Laplace's expansion and the adjugate matrix

The determinant of a {{math|2 × 2}} matrix is denoted either by "{{math|det}}" or by vertical bars around the matrix, and is defined as

:

For example,

:

The determinant has several key properties that can be proved by direct evaluation of the definition for $2\; \backslash times\; 2$-matrices, and that continue to hold for determinants of larger matrices. They are as follows:{{harvnb|Lang|1985|loc=§VII.1}} first, the determinant of the identity matrix is 1.

Second, the determinant is zero if two rows are the same:

:

This holds similarly if the two columns are the same. Moreover,

:

Finally, if any column is multiplied by some number $r$ (i.e., all entries in that column are multiplied by that number), the determinant is also multiplied by that number:

:

File:Area parallelogram as determinant modified.svg

If the matrix entries are real numbers, the matrix {{math|A}} represents the linear map that maps the basis vectors to the columns of {{math|A}}. The images of the basis vectors form a parallelogram that represents the image of the unit square under the mapping. The parallelogram defined by the columns of the above matrix is the one with vertices at {{math|(0, 0)}}, {{math|(a, c)}}, {{math|(a + b, c + d)}}, and {{math|(b, d)}}, as shown in the accompanying diagram.

The absolute value of {{math|ad − bc}} is the area of the parallelogram, and thus represents the scale factor by which areas are transformed by {{math|A}}.

The absolute value of the determinant together with the sign becomes the signed area of the parallelogram. The signed area is the same as the usual area, except that it is negative when the angle from the first to the second vector defining the parallelogram turns in a clockwise direction (which is opposite to the direction one would get for the identity matrix).

To show that {{math|ad − bc}} is the signed area, one may consider a matrix containing two vectors {{math|u ≡ (a, c)}} and {{math|v ≡ (b, d)}} representing the parallelogram's sides. The signed area can be expressed as {{math|{{!}}u{{!}} {{!}}v{{!}} sin θ}} for the angle θ between the vectors, which is simply base times height, the length of one vector times the perpendicular component of the other. Due to the sine this already is the signed area, yet it may be expressed more conveniently using the cosine of the complementary angle to a perpendicular vector, e.g. {{math|1=u^⊥ = (−c, a)}}, so that {{math|{{!}}u^⊥{{!}} {{!}}v{{!}} cos θ′}} becomes the signed area in question, which can be determined by the pattern of the scalar product to be equal to {{math|ad − bc}} according to the following equations:

: $\backslash text\{Signed\; area\}\; =$

|\boldsymbol{u}|\,|\boldsymbol{v}|\,\sin\,\theta = \left|\boldsymbol{u}^\perp\right|\,\left|\boldsymbol{v}\right|\,\cos\,\theta' =

\begin{pmatrix} -c \\ a \end{pmatrix} \cdot \begin{pmatrix} b \\ d \end{pmatrix} = ad - bc.

Thus the determinant gives the area scale factor and the orientation induced by the mapping represented by A. When the determinant is equal to one, the linear mapping defined by the matrix preserves area and orientation.

File:Determinant parallelepiped.svg is the absolute value of the determinant of the matrix formed by the columns constructed from the vectors r1, r2, and r3.]]

If an {{math|n × n}} real matrix A is written in terms of its column vectors , then

:$$

A\begin{pmatrix}1 \\ 0\\ \vdots \\0\end{pmatrix} = \mathbf{a}_1, \quad

A\begin{pmatrix}0 \\ 1\\ \vdots \\0\end{pmatrix} = \mathbf{a}_2, \quad

\ldots, \quad

A\begin{pmatrix}0 \\0 \\ \vdots \\1\end{pmatrix} = \mathbf{a}_n.

This means that $A$ maps the unit n-cube to the n-dimensional parallelotope defined by the vectors $\backslash mathbf\{a\}\_1,\; \backslash mathbf\{a\}\_2,\; \backslash ldots,\; \backslash mathbf\{a\}\_n,$ the region $P\; =\; \backslash left\backslash \{c\_1\; \backslash mathbf\{a\}\_1\; +\; \backslash cdots\; +\; c\_n\backslash mathbf\{a\}\_n\; \backslash mid\; 0\; \backslash leq\; c\_i\backslash leq\; 1\; \backslash \; \backslash forall\; i\backslash right\backslash \}$ ($\backslash forall$ stands for "for all" as a logical symbol.)

The determinant gives the signed n-dimensional volume of this parallelotope, $\backslash det(A)\; =\; \backslash pm\; \backslash text\{vol\}(P),$ and hence describes more generally the n-dimensional volume scale factor of the linear transformation produced by A.{{cite web|url=https://textbooks.math.gatech.edu/ila/determinants-volumes.html|title=Determinants and Volumes|website=textbooks.math.gatech.edu|access-date=16 March 2018}} (The sign shows whether the transformation preserves or reverses orientation.) In particular, if the determinant is zero, then this parallelotope has volume zero and is not fully n-dimensional, which indicates that the dimension of the image of A is less than n. This means that A produces a linear transformation which is neither onto nor one-to-one, and so is not invertible.

Let A be a square matrix with n rows and n columns, so that it can be written as

:$A\; =\; \backslash begin\{bmatrix\}$

a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\

a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\

\vdots & \vdots & \ddots & \vdots \\

a_{n,1} & a_{n,2} & \cdots & a_{n,n}

\end{bmatrix}.

The entries $a\_\{1,1\}$ etc. are, for many purposes, real or complex numbers. As discussed below, the determinant is also defined for matrices whose entries are in a commutative ring.

The determinant of A is denoted by det(A), or it can be denoted directly in terms of the matrix entries by writing enclosing bars instead of brackets:

:$\backslash begin\{vmatrix\}$

a_{1,1} & a_{1,2} & \cdots & a_{1,n} \\

a_{2,1} & a_{2,2} & \cdots & a_{2,n} \\

\vdots & \vdots & \ddots & \vdots \\

a_{n,1} & a_{n,2} & \cdots & a_{n,n}

\end{vmatrix}.

There are various equivalent ways to define the determinant of a square matrix A, i.e. one with the same number of rows and columns: the determinant can be defined via the Leibniz formula, an explicit formula involving sums of products of certain entries of the matrix. The determinant can also be characterized as the unique function depending on the entries of the matrix satisfying certain properties. This approach can also be used to compute determinants by simplifying the matrices in question.

{{main|Leibniz formula for determinants}}

The Leibniz formula for the determinant of a {{math|3 × 3}} matrix is the following:

:

= aei + bfg + cdh - ceg - bdi - afh.\

In this expression, each term has one factor from each row, all in different columns, arranged in increasing row order. For example, bdi has b from the first row second column, d from the second row first column, and i from the third row third column. The signs are determined by how many transpositions of factors are necessary to arrange the factors in increasing order of their columns (given that the terms are arranged left-to-right in increasing row order): positive for an even number of transpositions and negative for an odd number. For the example of bdi, the single transposition of bd to db gives dbi, whose three factors are from the first, second and third columns respectively; this is an odd number of transpositions, so the term appears with negative sign.

File:Sarrus rule1.svg]]

The rule of Sarrus is a mnemonic for the expanded form of this determinant: the sum of the products of three diagonal north-west to south-east lines of matrix elements, minus the sum of the products of three diagonal south-west to north-east lines of elements, when the copies of the first two columns of the matrix are written beside it as in the illustration. This scheme for calculating the determinant of a {{math|3 × 3}} matrix does not carry over into higher dimensions.

Generalizing the above to higher dimensions, the determinant of an $n\; \backslash times\; n$ matrix is an expression involving permutations and their signatures. A permutation of the set $\backslash \{1,\; 2,\; \backslash dots,\; n\; \backslash \}$ is a bijective function $\backslash sigma$ from this set to itself, with values $\backslash sigma(1),\; \backslash sigma(2),\backslash ldots,\backslash sigma(n)$ exhausting the entire set. The set of all such permutations, called the symmetric group, is commonly denoted $S\_n$. The signature $\backslash sgn(\backslash sigma)$ of a permutation $\backslash sigma$ is $+1,$ if the permutation can be obtained with an even number of transpositions (exchanges of two entries); otherwise, it is $-1.$

Given a matrix

:$A=\backslash begin\{bmatrix\}$

a_{1,1}\ldots a_{1,n}\\

\vdots\qquad\vdots\\

a_{n,1}\ldots a_{n,n}

\end{bmatrix},

the Leibniz formula for its determinant is, using sigma notation for the sum,

:$\backslash det(A)=\backslash begin\{vmatrix\}$

a_{1,1}\ldots a_{1,n}\\

\vdots\qquad\vdots\\

a_{n,1}\ldots a_{n,n}

\end{vmatrix} = \sum_{\sigma \in S_n}\sgn(\sigma)a_{1,\sigma(1)}\cdots a_{n,\sigma(n)}.

Using pi notation for the product, this can be shortened into

:$\backslash det(A)\; =\; \backslash sum\_\{\backslash sigma\; \backslash in\; S\_n\}\; \backslash left(\; \backslash sgn(\backslash sigma)\; \backslash prod\_\{i=1\}^n\; a\_\{i,\backslash sigma(i)\}\backslash right)$.

The Levi-Civita symbol $\backslash varepsilon\_\{i\_1,\backslash ldots,i\_n\}$ is defined on the {{mvar|n}}-tuples of integers in $\backslash \{1,\backslash ldots,n\backslash \}$ as {{math|0}} if two of the integers are equal, and otherwise as the signature of the permutation defined by the n-tuple of integers. With the Levi-Civita symbol, the Leibniz formula becomes

:$\backslash det(A)\; =\; \backslash sum\_\{i\_1,i\_2,\backslash ldots,i\_n\}\; \backslash varepsilon\_\{i\_1\backslash cdots\; i\_n\}\; a\_\{1,i\_1\}\; \backslash !\backslash cdots\; a\_\{n,i\_n\},$

where the sum is taken over all {{mvar|n}}-tuples of integers in $\backslash \{1,\backslash ldots,n\backslash \}.$

{{cite book |last1=McConnell |title=Applications of Tensor Analysis |url=https://archive.org/details/applicationoften0000mcco |url-access=registration |date=1957 |publisher=Dover Publications |pages=[https://archive.org/details/applicationoften0000mcco/page/10 10–17]}}{{harvnb|Harris|2014|loc=§4.7}}

The determinant can be characterized by the following three key properties. To state these, it is convenient to regard an $n\; \backslash times\; n$ matrix A as being composed of its $n$ columns, so denoted as

:$A\; =\; \backslash big\; (\; a\_1,\; \backslash dots,\; a\_n\; \backslash big\; ),$

where the column vector $a\_i$ (for each i) is composed of the entries of the matrix in the i-th column.

2. $\backslash det\backslash left(I\backslash right)\; =\; 1$, where $I$ is an identity matrix.
4. The determinant is multilinear: if the jth column of a matrix $A$ is written as a linear combination $a\_j\; =\; r\; \backslash cdot\; v\; +\; w$ of two column vectors v and w and a number r, then the determinant of A is expressible as a similar linear combination:
5. : $\backslash begin\{align\}|A|$

&= \big | a_1, \dots, a_{j-1}, r \cdot v + w, a_{j+1}, \dots, a_n | \\

&= r \cdot | a_1, \dots, v, \dots a_n | + | a_1, \dots, w, \dots, a_n |

\end{align}

2. The determinant is alternating: whenever two columns of a matrix are identical, its determinant is 0:
3. : $|\; a\_1,\; \backslash dots,\; v,\; \backslash dots,\; v,\; \backslash dots,\; a\_n|\; =\; 0.$

If the determinant is defined using the Leibniz formula as above, these three properties can be proved by direct inspection of that formula. Some authors also approach the determinant directly using these three properties: it can be shown that there is exactly one function that assigns to any $n\; \backslash times\; n$ matrix A a number that satisfies these three properties.Serge Lang, Linear Algebra, 2nd Edition, Addison-Wesley, 1971, pp 173, 191. This also shows that this more abstract approach to the determinant yields the same definition as the one using the Leibniz formula.

To see this it suffices to expand the determinant by multi-linearity in the columns into a (huge) linear combination of determinants of matrices in which each column is a standard basis vector. These determinants are either 0 (by property 9) or else ±1 (by properties 1 and 12 below), so the linear combination gives the expression above in terms of the Levi-Civita symbol. While less technical in appearance, this characterization cannot entirely replace the Leibniz formula in defining the determinant, since without it the existence of an appropriate function is not clear.{{citation needed|date=May 2021}}

These rules have several further consequences:

• The determinant is a homogeneous function, i.e., $$\backslash det(cA)\; =\; c^n\backslash det(A)$$ (for an $n\; \backslash times\; n$ matrix $A$).
• Interchanging any pair of columns of a matrix multiplies its determinant by −1. This follows from the determinant being multilinear and alternating (properties 2 and 3 above): $$|a\_1,\; \backslash dots,\; a\_j,\; \backslash dots\; a\_i,\; \backslash dots,\; a\_n|\; =\; -\; |a\_1,\; \backslash dots,\; a\_i,\; \backslash dots,\; a\_j,\; \backslash dots,\; a\_n|.$$ This formula can be applied iteratively when several columns are swapped. For example $$|a\_3,\; a\_1,\; a\_2,\; a\_4\; \backslash dots,\; a\_n|\; =\; -\; |a\_1,\; a\_3,\; a\_2,\; a\_4,\; \backslash dots,\; a\_n|\; =\; |a\_1,\; a\_2,\; a\_3,\; a\_4,\; \backslash dots,\; a\_n|.$$ Yet more generally, any permutation of the columns multiplies the determinant by the sign of the permutation.
• If some column can be expressed as a linear combination of the other columns (i.e. the columns of the matrix form a linearly dependent set), the determinant is 0. As a special case, this includes: if some column is such that all its entries are zero, then the determinant of that matrix is 0.
• Adding a scalar multiple of one column to another column does not change the value of the determinant. This is a consequence of multilinearity and being alternative: by multilinearity the determinant changes by a multiple of the determinant of a matrix with two equal columns, which determinant is 0, since the determinant is alternating.
• If $A$ is a triangular matrix, i.e. $a\_\{ij\}=0$, whenever $i>j$ or, alternatively, whenever

These characterizing properties and their consequences listed above are both theoretically significant, but can also be used to compute determinants for concrete matrices. In fact, Gaussian elimination can be applied to bring any matrix into upper triangular form, and the steps in this algorithm affect the determinant in a controlled way. The following concrete example illustrates the computation of the determinant of the matrix $A$ using that method:

:$A\; =\; \backslash begin\{bmatrix\}$

-2 & -1 & 2 \\

2 & 1 & 4 \\

-3 & 3 & -1

\end{bmatrix}.

Combining these equalities gives $|A|\; =\; -|E|\; =\; -(18\; \backslash cdot\; 3\; \backslash cdot\; (-1))\; =\; 54.$

The determinant of the transpose of $A$ equals the determinant of A:

:$\backslash det\backslash left(A^\backslash textsf\{T\}\backslash right)\; =\; \backslash det(A)$.

This can be proven by inspecting the Leibniz formula.{{harvnb|Lang|1987|loc=§VI.7, Theorem 7.5}} This implies that in all the properties mentioned above, the word "column" can be replaced by "row" throughout. For example, viewing an {{math|n × n}} matrix as being composed of n rows, the determinant is an n-linear function.

The determinant is a multiplicative map, i.e., for square matrices $A$ and $B$ of equal size, the determinant of a matrix product equals the product of their determinants:

:$\backslash det(AB)\; =\; \backslash det\; (A)\; \backslash det\; (B)$

This key fact can be proven by observing that, for a fixed matrix $B$, both sides of the equation are alternating and multilinear as a function depending on the columns of $A$. Moreover, they both take the value $\backslash det\; B$ when $A$ is the identity matrix. The above-mentioned unique characterization of alternating multilinear maps therefore shows this claim.Alternatively, {{harvnb|Bourbaki|1998|loc=§III.8, Proposition 1}} proves this result using the functoriality of the exterior power.

A matrix $A$ with entries in a field is invertible precisely if its determinant is nonzero. This follows from the multiplicativity of the determinant and the formula for the inverse involving the adjugate matrix mentioned below. In this event, the determinant of the inverse matrix is given by

:$\backslash det\backslash left(A^\{-1\}\backslash right)\; =\; \backslash frac\{1\}\{\backslash det(A)\}\; =\; [\backslash det(A)]^\{-1\}$.

In particular, products and inverses of matrices with non-zero determinant (respectively, determinant one) still have this property. Thus, the set of such matrices (of fixed size $n$ over a field $K$) forms a group known as the general linear group $\backslash operatorname\{GL\}\_n(K)$ (respectively, a subgroup called the special linear group $\backslash operatorname\{SL\}\_n(K)\; \backslash subset\; \backslash operatorname\{GL\}\_n(K)$. More generally, the word "special" indicates the subgroup of another matrix group of matrices of determinant one. Examples include the special orthogonal group (which if n is 2 or 3 consists of all rotation matrices), and the special unitary group.

Because the determinant respects multiplication and inverses, it is in fact a group homomorphism from $\backslash operatorname\{GL\}\_n(K)$ into the multiplicative group $K^\backslash times$ of nonzero elements of $K$. This homomorphism is surjective and its kernel is $\backslash operatorname\{SL\}\_n(K)$ (the matrices with determinant one). Hence, by the first isomorphism theorem, this shows that $\backslash operatorname\{SL\}\_n(K)$ is a normal subgroup of $\backslash operatorname\{GL\}\_n(K)$, and that the quotient group $\backslash operatorname\{GL\}\_n(K)/\backslash operatorname\{SL\}\_n(K)$ is isomorphic to $K^\backslash times$.

The Cauchy–Binet formula is a generalization of that product formula for rectangular matrices. This formula can also be recast as a multiplicative formula for compound matrices whose entries are the determinants of all quadratic submatrices of a given matrix.{{harvnb|Horn|Johnson|2018|loc=§0.8.7}}{{harvnb|Kung|Rota|Yan|2009|p=306}}

Laplace expansion expresses the determinant of a matrix $A$ recursively in terms of determinants of smaller matrices, known as its minors. The minor $M\_\{i,j\}$ is defined to be the determinant of the $(n-1)\; \backslash times\; (n-1)$ matrix that results from $A$ by removing the $i$-th row and the $j$-th column. The expression $(-1)^\{i+j\}M\_\{i,j\}$ is known as a cofactor. For every $i$, one has the equality

:$\backslash det(A)\; =\; \backslash sum\_\{j=1\}^n\; (-1)^\{i+j\}\; a\_\{i,j\}\; M\_\{i,j\},$

which is called the Laplace expansion along the {{mvar|i}}th row. For example, the Laplace expansion along the first row ($i=1$) gives the following formula:

:$$

\begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix} =

a\begin{vmatrix}e&f\\ h&i\end{vmatrix} - b\begin{vmatrix}d&f\\ g&i\end{vmatrix} + c\begin{vmatrix}d&e\\ g&h\end{vmatrix}

Unwinding the determinants of these $2\; \backslash times\; 2$-matrices gives back the Leibniz formula mentioned above. Similarly, the Laplace expansion along the $j$-th column is the equality

:$\backslash det(A)=\; \backslash sum\_\{i=1\}^n\; (-1)^\{i+j\}\; a\_\{i,j\}\; M\_\{i,j\}.$

Laplace expansion can be used iteratively for computing determinants, but this approach is inefficient for large matrices. However, it is useful for computing the determinants of highly symmetric matrix such as the Vandermonde matrix

$$\backslash begin\{vmatrix\}$$

1 & 1 & 1 & \cdots & 1 \\

x_1 & x_2 & x_3 & \cdots & x_n \\

x_1^2 & x_2^2 & x_3^2 & \cdots & x_n^2 \\

\vdots & \vdots & \vdots & \ddots & \vdots \\

x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \cdots & x_n^{n-1}

\end{vmatrix} =

\prod_{1 \leq i < j \leq n} \left(x_j - x_i\right).

The n-term Laplace expansion along a row or column can be generalized to write an n x n determinant as a sum of $\backslash tbinom\; nk$ terms, each the product of the determinant of a k x k submatrix and the determinant of the complementary (n−k) x (n−k) submatrix.

The adjugate matrix $\backslash operatorname\{adj\}(A)$ is the transpose of the matrix of the cofactors, that is,

: $(\backslash operatorname\{adj\}(A))\_\{i,j\}\; =\; (-1)^\{i+j\}\; M\_\{ji\}.$

For every matrix, one has{{harvnb|Horn|Johnson|2018|loc=§0.8.2}}.

: $(\backslash det\; A)\; I\; =\; A\backslash operatorname\{adj\}A\; =\; (\backslash operatorname\{adj\}A)\backslash ,A.$

Thus the adjugate matrix can be used for expressing the inverse of a nonsingular matrix:

: $A^\{-1\}\; =\; \backslash frac\; 1\{\backslash det\; A\}\backslash operatorname\{adj\}A.$

The formula for the determinant of a $2\; \backslash times\; 2$ matrix above continues to hold, under appropriate further assumptions, for a block matrix, i.e., a matrix composed of four submatrices $A,\; B,\; C,\; D$ of dimension $m\; \backslash times\; m$, $m\; \backslash times\; n$, $n\; \backslash times\; m$ and $n\; \backslash times\; n$, respectively. The easiest such formula, which can be proven using either the Leibniz formula or a factorization involving the Schur complement, is

:

If $A$ is invertible, then it follows with results from the section on multiplicativity that

:$\backslash begin\{align\}$

\det\begin{pmatrix}A& B\\ C& D\end{pmatrix}

& = \det(A)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix}

\underbrace{\det\begin{pmatrix}A^{-1}& -A^{-1} B\\ 0& I_n\end{pmatrix}}_{=\,\det(A^{-1})\,=\,(\det A)^{-1}}\\

& = \det(A) \det\begin{pmatrix}I_m& 0\\ C A^{-1}& D-C A^{-1} B\end{pmatrix}\\

& = \det(A) \det(D - C A^{-1} B),

\end{align}

which simplifies to $\backslash det\; (A)\; (D\; -\; C\; A^\{-1\}\; B)$ when $D$ is a $1\; \backslash times\; 1$ matrix.

A similar result holds when $D$ is invertible, namely

:$\backslash begin\{align\}$

\det\begin{pmatrix}A& B\\ C& D\end{pmatrix}

& = \det(D)\det\begin{pmatrix}A& B\\ C& D\end{pmatrix}

\underbrace{\det\begin{pmatrix}I_m& 0\\ -D^{-1} C& D^{-1}\end{pmatrix}}_{=\,\det(D^{-1})\,=\,(\det D)^{-1}}\\

& = \det(D) \det\begin{pmatrix}A - B D^{-1} C& B D^{-1}\\ 0& I_n\end{pmatrix}\\

& = \det(D) \det(A - B D^{-1} C).

\end{align}

Both results can be combined to derive Sylvester's determinant theorem, which is also stated below.

If the blocks are square matrices of the same size further formulas hold. For example, if $C$ and $D$ commute (i.e., $CD=DC$), then{{Cite journal|first=J. R.|last= Silvester|title= Determinants of Block Matrices|journal= Math. Gaz.|volume=84 |issue= 501|year=2000 | pages= 460–467| jstor=3620776|url= https://hal.archives-ouvertes.fr/hal-01509379/document|doi= 10.2307/3620776|s2cid= 41879675}}

:

This formula has been generalized to matrices composed of more than $2\; \backslash times\; 2$ blocks, again under appropriate commutativity conditions among the individual blocks.{{cite journal|last1=Sothanaphan|first1=Nat|title=Determinants of block matrices with noncommuting blocks|journal=Linear Algebra and Its Applications|date=January 2017|volume=512| pages=202–218| doi=10.1016/j.laa.2016.10.004|arxiv=1805.06027|s2cid=119272194}}

For $A\; =\; D$ and $B\; =\; C$, the following formula holds (even if $A$ and $B$ do not commute).

:

Sylvester's determinant theorem states that for A, an {{math|m × n}} matrix, and B, an {{math|n × m}} matrix (so that A and B have dimensions allowing them to be multiplied in either order forming a square matrix):

:$\backslash det\backslash left(I\_\backslash mathit\{m\}\; +\; AB\backslash right)\; =\; \backslash det\backslash left(I\_\backslash mathit\{n\}\; +\; BA\backslash right),$

where I_m and I_n are the {{math|m × m}} and {{math|n × n}} identity matrices, respectively.

From this general result several consequences follow.

{{ordered list

| list-style-type=lower-alpha

| For the case of column vector c and row vector r, each with m components, the formula allows quick calculation of the determinant of a matrix that differs from the identity matrix by a matrix of rank 1:

:$\backslash det\backslash left(I\_\backslash mathit\{m\}\; +\; cr\backslash right)\; =\; 1\; +\; rc.$

| More generally,Proofs can be found in http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/proof003.html for any invertible {{math|m × m}} matrix X,

:$\backslash det(X\; +\; AB)\; =\; \backslash det(X)\; \backslash det\backslash left(I\_\backslash mathit\{n\}\; +\; BX^\{-1\}A\backslash right),$

| For a column and row vector as above:

: $\backslash det(X\; +\; cr)\; =\; \backslash det(X)\; \backslash det\backslash left(1\; +\; rX^\{-1\}c\backslash right)\; =\; \backslash det(X)\; +\; r\backslash ,\backslash operatorname\{adj\}(X)\backslash ,c.$

| For square matrices $A$ and $B$ of the same size, the matrices $AB$ and $BA$ have the same characteristic polynomials (hence the same eigenvalues).

}}

A generalization is $\backslash det\backslash left(Z\; +\; AWB\backslash right)\; =\; \backslash det\backslash left(\; Z\backslash right)\; \backslash det\backslash left(W\; \backslash right)\; \backslash det\backslash left(W^\{-1\}\; +\; B\; Z^\{-1\}\; A\backslash right)$(see Matrix determinant lemma), where Z is an {{math|m × m}} invertible matrix and W is an {{math|n × n}} invertible matrix.

The determinant of the sum $A+B$ of two square matrices of the same size is not in general expressible in terms of the determinants of A and of B.

However, for positive semidefinite matrices $A$, $B$ and $C$ of equal size,

$$\backslash det(A\; +\; B\; +\; C)\; +\; \backslash det(C)\; \backslash geq\; \backslash det(A\; +\; C)\; +\; \backslash det(B\; +\; C)\backslash text\{,\}$$

with the corollary{{cite arXiv| last1=Lin| first1=Minghua| last2=Sra|first2=Suvrit|title=Completely strong superadditivity of generalized matrix functions|eprint=1410.1958| class=math.FA| year=2014}}{{cite journal|last1=Paksoy|last2=Turkmen|last3=Zhang|title=Inequalities of Generalized Matrix Functions via Tensor Products|journal=Electronic Journal of Linear Algebra|year=2014|volume=27|pages= 332–341| doi=10.13001/1081-3810.1622|url=https://nsuworks.nova.edu/cgi/viewcontent.cgi?article=1062&context=math_facarticles|doi-access=free}}

$$\backslash det(A\; +\; B)\; \backslash geq\; \backslash det(A)\; +\; \backslash det(B)\backslash text\{.\}$$

Brunn–Minkowski theorem implies that the {{mvar|n}}th root of determinant is a concave function, when restricted to Hermitian positive-definite $n\backslash times\; n$ matrices.{{cite web|url=https://mathoverflow.net/questions/42594/concavity-of-det1-n-over-hpd-n|title=Concavity of det^1/_n over HPD_n.|date=Oct 18, 2010|last1=Serre|first1=Denis|website=MathOverflow}} Therefore, if {{mvar|A}} and {{mvar|B}} are Hermitian positive-definite $n\backslash times\; n$ matrices, one has

$$\backslash sqrt[n]\{\backslash det(A+B)\}\backslash geq\backslash sqrt[n]\{\backslash det(A)\}+\backslash sqrt[n]\{\backslash det(B)\},$$ since the {{mvar|n}}th root of the determinant is a homogeneous function.

For the special case of $2\backslash times\; 2$ matrices with complex entries, the determinant of the sum can be written in terms of determinants and traces in the following identity:

:$\backslash det(A+B)\; =\; \backslash det(A)\; +\; \backslash det(B)\; +\; \backslash text\{tr\}(A)\backslash text\{tr\}(B)\; -\; \backslash text\{tr\}(AB).$

The determinant is closely related to two other central concepts in linear algebra, the eigenvalues and the characteristic polynomial of a matrix. Let $A$ be an $n\; \backslash times\; n$ matrix with complex entries. Then, by the Fundamental Theorem of Algebra, $A$ must have exactly n eigenvalues $\backslash lambda\_1,\; \backslash lambda\_2,\; \backslash ldots,\; \backslash lambda\_n$. (Here it is understood that an eigenvalue with algebraic multiplicity {{mvar|μ}} occurs {{mvar|μ}} times in this list.) Then, it turns out the determinant of {{mvar|A}} is equal to the product of these eigenvalues,

:$\backslash det(A)\; =\; \backslash prod\_\{i=1\}^n\; \backslash lambda\_i=\backslash lambda\_1\backslash lambda\_2\backslash cdots\backslash lambda\_n.$

The product of all non-zero eigenvalues is referred to as pseudo-determinant.

From this, one immediately sees that the determinant of a matrix $A$ is zero if and only if $0$ is an eigenvalue of $A$. In other words, $A$ is invertible if and only if $0$ is not an eigenvalue of $A$.

The characteristic polynomial is defined as{{harvnb|Lang|1985|loc=§VIII.2}}, {{harvnb|Horn|Johnson|2018|loc=Def. 1.2.3}}

:$\backslash chi\_A(t)\; =\; \backslash det(t\; \backslash cdot\; I\; -\; A).$

Here, $t$ is the indeterminate of the polynomial and $I$ is the identity matrix of the same size as $A$. By means of this polynomial, determinants can be used to find the eigenvalues of the matrix $A$: they are precisely the roots of this polynomial, i.e., those complex numbers $\backslash lambda$ such that

:$\backslash chi\_A(\backslash lambda)\; =\; 0.$

A Hermitian matrix is positive definite if all its eigenvalues are positive. Sylvester's criterion asserts that this is equivalent to the determinants of the submatrices

:$A\_k\; :=\; \backslash begin\{bmatrix\}$

a_{1,1} & a_{1,2} & \cdots & a_{1,k} \\

a_{2,1} & a_{2,2} & \cdots & a_{2,k} \\

\vdots & \vdots & \ddots & \vdots \\

a_{k,1} & a_{k,2} & \cdots & a_{k,k}

\end{bmatrix}

being positive, for all $k$ between $1$ and $n$.{{harvnb|Horn|Johnson|2018|loc=Observation 7.1.2, Theorem 7.2.5}}

The trace tr(A) is by definition the sum of the diagonal entries of {{mvar|A}} and also equals the sum of the eigenvalues. Thus, for complex matrices {{mvar|A}},

:$\backslash det(\backslash exp(A))\; =\; \backslash exp(\backslash operatorname\{tr\}(A))$

or, for real matrices {{mvar|A}},

:$\backslash operatorname\{tr\}(A)\; =\; \backslash log(\backslash det(\backslash exp(A))).$

Here exp({{mvar|A}}) denotes the matrix exponential of {{mvar|A}}, because every eigenvalue {{mvar|λ}} of {{mvar|A}} corresponds to the eigenvalue exp({{mvar|λ}}) of exp({{mvar|A}}). In particular, given any logarithm of {{mvar|A}}, that is, any matrix {{mvar|L}} satisfying

:$\backslash exp(L)\; =\; A$

the determinant of {{mvar|A}} is given by

:$\backslash det(A)\; =\; \backslash exp(\backslash operatorname\{tr\}(L)).$

For example, for {{math|1=n = 2}}, {{math|1=n = 3}}, and {{math|1=n = 4}}, respectively,

:$\backslash begin\{align\}$

\det(A) &= \frac{1}{2}\left(\left(\operatorname{tr}(A)\right)^2 - \operatorname{tr}\left(A^2\right)\right), \\

\det(A) &= \frac{1}{6}\left(\left(\operatorname{tr}(A)\right)^3 - 3\operatorname{tr}(A) ~ \operatorname{tr}\left(A^2\right) + 2 \operatorname{tr}\left(A^3\right)\right), \\

\det(A) &= \frac{1}{24}\left(\left(\operatorname{tr}(A)\right)^4 - 6\operatorname{tr}\left(A^2\right)\left(\operatorname{tr}(A)\right)^2 + 3\left(\operatorname{tr}\left(A^2\right)\right)^2 + 8\operatorname{tr}\left(A^3\right)~\operatorname{tr}(A) - 6\operatorname{tr}\left(A^4\right)\right).

\end{align}

cf. Cayley-Hamilton theorem. Such expressions are deducible from combinatorial arguments, Newton's identities, or the Faddeev–LeVerrier algorithm. That is, for generic {{mvar|n}}, {{math|detA {{=}} (−1)ⁿc₀}} the signed constant term of the characteristic polynomial, determined recursively from

:$c\_n\; =\; 1;\; ~~~c\_\{n-m\}\; =\; -\backslash frac\{1\}\{m\}\backslash sum\_\{k=1\}^m\; c\_\{n-m+k\}\; \backslash operatorname\{tr\}\backslash left(A^k\backslash right)\; ~~(1\; \backslash le\; m\; \backslash le\; n)~.$

In the general case, this may also be obtained fromA proof can be found in the Appendix B of {{cite journal | last1 = Kondratyuk | first1 = L. A. | last2 = Krivoruchenko | first2 = M. I. | year = 1992 | title = Superconducting quark matter in SU(2) color group | journal = Zeitschrift für Physik A | volume = 344 | issue = 1| pages = 99–115 | doi = 10.1007/BF01291027 | bibcode = 1992ZPhyA.344...99K | s2cid = 120467300 }}

:$\backslash det(A)\; =\; \backslash sum\_\{\backslash begin\{array\}\{c\}k\_1,k\_2,\backslash ldots,k\_n\; \backslash geq\; 0\backslash \backslash k\_1+2k\_2+\backslash cdots+nk\_n=n\backslash end\{array\}\}\backslash prod\_\{l=1\}^n\; \backslash frac\{(-1)^\{k\_l+1\}\}\{l^\{k\_l\}k\_l!\}\; \backslash operatorname\{tr\}\backslash left(A^l\backslash right)^\{k\_l\},$

where the sum is taken over the set of all integers {{math|k_l ≥ 0}} satisfying the equation

:$\backslash sum\_\{l=1\}^n\; lk\_l\; =\; n.$

The formula can be expressed in terms of the complete exponential Bell polynomial of n arguments s_l = −(l – 1)! tr(A^l) as

:$\backslash det(A)\; =\; \backslash frac\{(-1)^n\}\{n!\}\; B\_n(s\_1,\; s\_2,\; \backslash ldots,\; s\_n).$

This formula can also be used to find the determinant of a matrix {{math|A^I_J}} with multidimensional indices {{math|1=I = (i₁, i₂, ..., i_r)}} and {{math|1=J = (j₁, j₂, ..., j_r)}}. The product and trace of such matrices are defined in a natural way as

:$(AB)^I\_J\; =\; \backslash sum\_K\; A^I\_K\; B^K\_J,\; \backslash operatorname\{tr\}(A)\; =\; \backslash sum\_I\; A^I\_I.$

An important arbitrary dimension {{mvar|n}} identity can be obtained from the Mercator series expansion of the logarithm when the expansion converges. If every eigenvalue of A is less than 1 in absolute value,

:$\backslash det(I\; +\; A)\; =\; \backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{k!\}\; \backslash left(-\backslash sum\_\{j=1\}^\backslash infty\; \backslash frac\{(-1)^j\}\{j\}\; \backslash operatorname\{tr\}\backslash left(A^j\backslash right)\backslash right)^k\backslash ,,$

where {{math|I}} is the identity matrix. More generally, if

:$\backslash sum\_\{k=0\}^\backslash infty\; \backslash frac\{1\}\{k!\}\; \backslash left(-\backslash sum\_\{j=1\}^\backslash infty\; \backslash frac\{(-1)^j\; s^j\}\{j\}\backslash operatorname\{tr\}\backslash left(A^j\backslash right)\backslash right)^k\backslash ,,$

is expanded as a formal power series in {{mvar|s}} then all coefficients of {{mvar|s}}^{{mvar|m}} for {{math|m > n}} are zero and the remaining polynomial is {{math|det(I + sA)}}.

For a positive definite matrix {{math|A}}, the trace operator gives the following tight lower and upper bounds on the log determinant

:$\backslash operatorname\{tr\}\backslash left(I\; -\; A^\{-1\}\backslash right)\; \backslash le\; \backslash log\backslash det(A)\; \backslash le\; \backslash operatorname\{tr\}(A\; -\; I)$

with equality if and only if {{math|1=A = I}}. This relationship can be derived via the formula for the Kullback-Leibler divergence between two multivariate normal distributions.

Also,

:$\backslash frac\{n\}\{\backslash operatorname\{tr\}\backslash left(A^\{-1\}\backslash right)\}\; \backslash leq\; \backslash det(A)^\backslash frac\{1\}\{n\}\; \backslash leq\; \backslash frac\{1\}\{n\}\backslash operatorname\{tr\}(A)\; \backslash leq\; \backslash sqrt\{\backslash frac\{1\}\{n\}\backslash operatorname\{tr\}\backslash left(A^2\backslash right)\}.$

These inequalities can be proved by expressing the traces and the determinant in terms of the eigenvalues. As such, they represent the well-known fact that the harmonic mean is less than the geometric mean, which is less than the arithmetic mean, which is, in turn, less than the root mean square.

The Leibniz formula shows that the determinant of real (or analogously for complex) square matrices is a polynomial function from $\backslash mathbf\; R^\{n\; \backslash times\; n\}$ to $\backslash mathbf\; R$. In particular, it is everywhere differentiable. Its derivative can be expressed using Jacobi's formula:{{harvnb|Horn|Johnson|2018|loc=§ 0.8.10}}

:$\backslash frac\{d\; \backslash det(A)\}\{d\; \backslash alpha\}\; =\; \backslash operatorname\{tr\}\backslash left(\backslash operatorname\{adj\}(A)\; \backslash frac\{d\; A\}\{d\; \backslash alpha\}\backslash right).$

where $\backslash operatorname\{adj\}(A)$ denotes the adjugate of $A$. In particular, if $A$ is invertible, we have

:$\backslash frac\{d\; \backslash det(A)\}\{d\; \backslash alpha\}\; =\; \backslash det(A)\; \backslash operatorname\{tr\}\backslash left(A^\{-1\}\; \backslash frac\{d\; A\}\{d\; \backslash alpha\}\backslash right).$

Expressed in terms of the entries of $A$, these are

: $\backslash frac\{\backslash partial\; \backslash det(A)\}\{\backslash partial\; A\_\{ij\}\}=\; \backslash operatorname\{adj\}(A)\_\{ji\}\; =\; \backslash det(A)\backslash left(A^\{-1\}\backslash right)\_\{ji\}.$

Yet another equivalent formulation is

:$\backslash det(A\; +\; \backslash epsilon\; X)\; -\; \backslash det(A)\; =\; \backslash operatorname\{tr\}(\backslash operatorname\{adj\}(A)\; X)\; \backslash epsilon\; +\; O\backslash left(\backslash epsilon^2\backslash right)\; =\; \backslash det(A)\; \backslash operatorname\{tr\}\backslash left(A^\{-1\}\; X\backslash right)\; \backslash epsilon\; +\; O\backslash left(\backslash epsilon^2\backslash right)$,

using big O notation. The special case where $A\; =\; I$, the identity matrix, yields

:$\backslash det(I\; +\; \backslash epsilon\; X)\; =\; 1\; +\; \backslash operatorname\{tr\}(X)\; \backslash epsilon\; +\; O\backslash left(\backslash epsilon^2\backslash right).$

This identity is used in describing Lie algebras associated to certain matrix Lie groups. For example, the special linear group $\backslash operatorname\{SL\}\_n$ is defined by the equation $\backslash det\; A\; =\; 1$. The above formula shows that its Lie algebra is the special linear Lie algebra $\backslash mathfrak\{sl\}\_n$ consisting of those matrices having trace zero.

Writing a $3\; \backslash times\; 3$ matrix as where $a,\; b,c$ are column vectors of length 3, then the gradient over one of the three vectors may be written as the cross product of the other two:

: $\backslash begin\{align\}$

\nabla_\mathbf{a}\det(A) &= \mathbf{b} \times \mathbf{c} \\

\nabla_\mathbf{b}\det(A) &= \mathbf{c} \times \mathbf{a} \\

\nabla_\mathbf{c}\det(A) &= \mathbf{a} \times \mathbf{b}.

\end{align}

Historically, determinants were used long before matrices: A determinant was originally defined as a property of a system of linear equations.

The determinant "determines" whether the system has a unique solution (which occurs precisely if the determinant is non-zero).

In this sense, determinants were first used in the Chinese mathematics textbook The Nine Chapters on the Mathematical Art (九章算術, Chinese scholars, around the 3rd century BCE). In Europe, solutions of linear systems of two equations were expressed by Cardano in 1545 by a determinant-like entity.{{harvnb|Grattan-Guinness|2003|loc=§6.6}}

Determinants proper originated separately from the work of Seki Takakazu in 1683 in Japan and parallelly of Leibniz in 1693.Cajori, F. [https://archive.org/details/ahistorymathema02cajogoog/page/n94 A History of Mathematics p. 80]{{harvnb|Eves|1990|p=405}}A Brief History of Linear Algebra and Matrix Theory at: {{cite web |url=http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html |title=A Brief History of Linear Algebra and Matrix Theory |access-date=2012-01-24 |url-status=dead |archive-url=https://web.archive.org/web/20120910034016/http://darkwing.uoregon.edu/~vitulli/441.sp04/LinAlgHistory.html |archive-date=2012-09-10 |df=dmy-all}} {{harvtxt|Cramer|1750}} stated, without proof, Cramer's rule.{{harvnb|Kleiner|2007|p=80}} Both Cramer and also {{harvtxt|Bézout|1779}} were led to determinants by the question of plane curves passing through a given set of points.{{harvtxt|Bourbaki|1994|p=59}}

Vandermonde (1771) first recognized determinants as independent functions.Campbell, H: "Linear Algebra With Applications", pages 111–112. Appleton Century Crofts, 1971 {{harvtxt|Laplace|1772}} gave the general method of expanding a determinant in terms of its complementary minors: Vandermonde had already given a special case.Muir, Sir Thomas, The Theory of Determinants in the historical Order of Development [London, England: Macmillan and Co., Ltd., 1906]. {{JFM|37.0181.02}} Immediately following, Lagrange (1773) treated determinants of the second and third order and applied it to questions of elimination theory; he proved many special cases of general identities.

Gauss (1801) made the next advance. Like Lagrange, he made much use of determinants in the theory of numbers. He introduced the word "determinant" (Laplace had used "resultant"), though not in the present signification, but rather as applied to the discriminant of a quadratic form.{{harvnb|Kleiner|2007|loc=§5.2}} Gauss also arrived at the notion of reciprocal (inverse) determinants, and came very near the multiplication theorem.{{Clarify|date=June 2023|reason=What is "the multiplication theorem"?}}

The next contributor of importance is Binet (1811, 1812), who formally stated the theorem relating to the product of two matrices of m columns and n rows, which for the special case of {{math|1=m = n}} reduces to the multiplication theorem. On the same day (November 30, 1812) that Binet presented his paper to the Academy, Cauchy also presented one on the subject. (See Cauchy–Binet formula.) In this he used the word "determinant" in its present sense,The first use of the word "determinant" in the modern sense appeared in: Cauchy, Augustin-Louis "Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs égales et des signes contraires par suite des transpositions operées entre les variables qu'elles renferment," which was first read at the Institute de France in Paris on November 30, 1812, and which was subsequently published in the Journal de l'Ecole Polytechnique, Cahier 17, Tome 10, pages 29–112 (1815).Origins of mathematical terms: http://jeff560.tripod.com/d.html summarized and simplified what was then known on the subject, improved the notation, and gave the multiplication theorem with a proof more satisfactory than Binet's.History of matrices and determinants: http://www-history.mcs.st-and.ac.uk/history/HistTopics/Matrices_and_determinants.html With him begins the theory in its generality.

{{harvtxt|Jacobi|1841}} used the functional determinant which Sylvester later called the Jacobian.{{harvnb|Eves|1990|p=494}} In his memoirs in Crelle's Journal for 1841 he specially treats this subject, as well as the class of alternating functions which Sylvester has called alternants. About the time of Jacobi's last memoirs, Sylvester (1839) and Cayley began their work. {{harvnb|Cayley|1841}} introduced the modern notation for the determinant using vertical bars.{{harvnb|Cajori|1993|loc=Vol. II, p. 92, no. 462}}History of matrix notation: http://jeff560.tripod.com/matrices.html

The study of special forms of determinants has been the natural result of the completion of the general theory. Axisymmetric determinants have been studied by Lebesgue, Hesse, and Sylvester; persymmetric determinants by Sylvester and Hankel; circulants by Catalan, Spottiswoode, Glaisher, and Scott; skew determinants and Pfaffians, in connection with the theory of orthogonal transformation, by Cayley; continuants by Sylvester; Wronskians (so called by Muir) by Christoffel and Frobenius; compound determinants by Sylvester, Reiss, and Picquet; Jacobians and Hessians by Sylvester; and symmetric gauche determinants by Trudi. Of the textbooks on the subject Spottiswoode's was the first. In America, Hanus (1886), Weld (1893), and Muir/Metzler (1933) published treatises.

Determinants can be used to describe the solutions of a linear system of equations, written in matrix form as $Ax\; =\; b$. This equation has a unique solution $x$ if and only if $\backslash det\; (A)$ is nonzero. In this case, the solution is given by Cramer's rule:

:$x\_i\; =\; \backslash frac\{\backslash det(A\_i)\}\{\backslash det(A)\}\; \backslash qquad\; i\; =\; 1,\; 2,\; 3,\; \backslash ldots,\; n$

where $A\_i$ is the matrix formed by replacing the $i$-th column of $A$ by the column vector $b$. This follows immediately by column expansion of the determinant, i.e.

:$\backslash det(A\_i)\; =$

\det\begin{bmatrix}a_1 & \ldots & b & \ldots & a_n\end{bmatrix}

=\sum_{j=1}^n x_j\det\begin{bmatrix}a_1 & \ldots & a_{i-1} & a_j & a_{i+1} & \ldots & a_n\end{bmatrix} =

x_i\det(A)

where the vectors $a\_j$ are the columns of A. The rule is also implied by the identity

:$A\backslash ,\; \backslash operatorname\{adj\}(A)\; =\; \backslash operatorname\{adj\}(A)\backslash ,\; A\; =\; \backslash det(A)\backslash ,\; I\_n.$

Cramer's rule can be implemented in $\backslash operatorname\; O(n^3)$ time, which is comparable to more common methods of solving systems of linear equations, such as LU, QR, or singular value decomposition.{{harvnb|Habgood|Arel|2012}}

Determinants can be used to characterize linearly dependent vectors: $\backslash det\; A$ is zero if and only if the column vectors of the matrix $A$ are linearly dependent.{{harvnb|Lang|1985|loc=§VII.3}} For example, given two linearly independent vectors $v\_1,\; v\_2\; \backslash in\; \backslash mathbf\; R^3$, a third vector $v\_3$ lies in the plane spanned by the former two vectors exactly if the determinant of the $3\; \backslash times\; 3$ matrix consisting of the three vectors is zero. The same idea is also used in the theory of differential equations: given functions $f\_1(x),\; \backslash dots,\; f\_n(x)$ (supposed to be $n-1$ times differentiable), the Wronskian is defined to be

:$W(f\_1,\; \backslash ldots,\; f\_n)(x)\; =$

\begin{vmatrix}

f_1(x) & f_2(x) & \cdots & f_n(x) \\

f_1'(x) & f_2'(x) & \cdots & f_n'(x) \\

\vdots & \vdots & \ddots & \vdots \\

f_1^{(n-1)}(x) & f_2^{(n-1)}(x) & \cdots & f_n^{(n-1)}(x)

\end{vmatrix}.

It is non-zero (for some $x$) in a specified interval if and only if the given functions and all their derivatives up to order $n-1$ are linearly independent. If it can be shown that the Wronskian is zero everywhere on an interval then, in the case of analytic functions, this implies the given functions are linearly dependent. See the Wronskian and linear independence. Another such use of the determinant is the resultant, which gives a criterion when two polynomials have a common root.{{harvnb|Lang|2002|loc=§IV.8}}

{{Main|Orientation (vector space)}}

The determinant can be thought of as assigning a number to every sequence of n vectors in Rⁿ, by using the square matrix whose columns are the given vectors. The determinant will be nonzero if and only if the sequence of vectors is a basis for Rⁿ. In that case, the sign of the determinant determines whether the orientation of the basis is consistent with or opposite to the orientation of the standard basis. In the case of an orthogonal basis, the magnitude of the determinant is equal to the product of the lengths of the basis vectors. For instance, an orthogonal matrix with entries in Rⁿ represents an orthonormal basis in Euclidean space, and hence has determinant of ±1 (since all the vectors have length 1). The determinant is +1 if and only if the basis has the same orientation. It is −1 if and only if the basis has the opposite orientation.

More generally, if the determinant of A is positive, A represents an orientation-preserving linear transformation (if A is an orthogonal {{math|2 × 2}} or {{math|3 × 3}} matrix, this is a rotation), while if it is negative, A switches the orientation of the basis.

As pointed out above, the absolute value of the determinant of real vectors is equal to the volume of the parallelepiped spanned by those vectors. As a consequence, if $f\; :\; \backslash mathbf\; R^n\; \backslash to\; \backslash mathbf\; R^n$ is the linear map given by multiplication with a matrix $A$, and $S\; \backslash subset\; \backslash mathbf\; R^n$ is any measurable subset, then the volume of $f(S)$ is given by $|\backslash det(A)|$ times the volume of $S$.{{harvnb|Lang|1985|loc=§VII.6, Theorem 6.10}} More generally, if the linear map $f\; :\; \backslash mathbf\; R^n\; \backslash to\; \backslash mathbf\; R^m$ is represented by the $m\; \backslash times\; n$ matrix $A$, then the $n$-dimensional volume of $f(S)$ is given by:

:$\backslash operatorname\{volume\}(f(S))\; =\; \backslash sqrt\{\backslash det\backslash left(A^\backslash textsf\{T\}\; A\backslash right)\}\; \backslash operatorname\{volume\}(S).$

By calculating the volume of the tetrahedron bounded by four points, they can be used to identify skew lines. The volume of any tetrahedron, given its vertices $a,\; b,\; c,\; d$, $\backslash frac\; 1\; 6\; \backslash cdot\; |\backslash det(a-b,b-c,c-d)|$, or any other combination of pairs of vertices that form a spanning tree over the vertices.

File:Jacobian_determinant_and_distortion.svg

For a general differentiable function, much of the above carries over by considering the Jacobian matrix of f. For

:$f:\; \backslash mathbf\; R^n\; \backslash rightarrow\; \backslash mathbf\; R^n,$

the Jacobian matrix is the {{math|n × n}} matrix whose entries are given by the partial derivatives

:$D(f)\; =\; \backslash left(\backslash frac\; \{\backslash partial\; f\_i\}\{\backslash partial\; x\_j\}\backslash right)\_\{1\; \backslash leq\; i,\; j\; \backslash leq\; n\}.$

Its determinant, the Jacobian determinant, appears in the higher-dimensional version of integration by substitution: for suitable functions f and an open subset U of Rⁿ (the domain of f), the integral over f(U) of some other function {{math|φ : Rⁿ → R^m}} is given by

:$\backslash int\_\{f(U)\}\; \backslash phi(\backslash mathbf\{v\})\backslash ,\; d\backslash mathbf\{v\}\; =\; \backslash int\_U\; \backslash phi(f(\backslash mathbf\{u\}))\; \backslash left|\backslash det(\backslash operatorname\{D\}f)(\backslash mathbf\{u\})\backslash right|\; \backslash ,d\backslash mathbf\{u\}.$

The Jacobian also occurs in the inverse function theorem.

When applied to the field of Cartography, the determinant can be used to measure the rate of expansion of a map near the poles.{{Cite book|last=Lay|first=David|title=Linear Algebra and Its Applications 6th Edition|publisher=Pearson|year=2021|pages=172|language=English}}

The above identities concerning the determinant of products and inverses of matrices imply that similar matrices have the same determinant: two matrices A and B are similar, if there exists an invertible matrix X such that {{math|1=A = X⁻¹BX}}. Indeed, repeatedly applying the above identities yields

:$\backslash det(A)\; =\; \backslash det(X)^\{-1\}\; \backslash det(B)\backslash det(X)\; =\; \backslash det(B)\; \backslash det(X)^\{-1\}\; \backslash det(X)\; =\; \backslash det(B).$

The determinant is therefore also called a similarity invariant. The determinant of a linear transformation

:$T\; :\; V\; \backslash to\; V$

for some finite-dimensional vector space V is defined to be the determinant of the matrix describing it, with respect to an arbitrary choice of basis in V. By the similarity invariance, this determinant is independent of the choice of the basis for V and therefore only depends on the endomorphism T.

The above definition of the determinant using the Leibniz rule holds works more generally when the entries of the matrix are elements of a commutative ring $R$, such as the integers $\backslash mathbf\; Z$, as opposed to the field of real or complex numbers. Moreover, the characterization of the determinant as the unique alternating multilinear map that satisfies $\backslash det(I)\; =\; 1$ still holds, as do all the properties that result from that characterization.{{harvnb|Dummit|Foote|2004|loc=§11.4}}

A matrix $A\; \backslash in\; \backslash operatorname\{Mat\}\_\{n\; \backslash times\; n\}(R)$ is invertible (in the sense that there is an inverse matrix whose entries are in $R$) if and only if its determinant is an invertible element in $R$.{{harvnb|Dummit|Foote|2004|loc=§11.4, Theorem 30}} For $R\; =\; \backslash mathbf\; Z$, this means that the determinant is +1 or −1. Such a matrix is called unimodular.

The determinant being multiplicative, it defines a group homomorphism

:$\backslash operatorname\{GL\}\_n(R)\; \backslash rightarrow\; R^\backslash times,$

between the general linear group (the group of invertible $n\; \backslash times\; n$-matrices with entries in $R$) and the multiplicative group of units in $R$. Since it respects the multiplication in both groups, this map is a group homomorphism.

Image:Determinant as a natural transformation.svg

Given a ring homomorphism $f\; :\; R\; \backslash to\; S$, there is a map $\backslash operatorname\{GL\}\_n(f)\; :\; \backslash operatorname\{GL\}\_n(R)\; \backslash to\; \backslash operatorname\{GL\}\_n(S)$ given by replacing all entries in $R$ by their images under $f$. The determinant respects these maps, i.e., the identity

:$f(\backslash det((a\_\{i,j\})))\; =\; \backslash det\; ((f(a\_\{i,j\})))$

holds. In other words, the displayed commutative diagram commutes.

For example, the determinant of the complex conjugate of a complex matrix (which is also the determinant of its conjugate transpose) is the complex conjugate of its determinant, and for integer matrices: the reduction modulo $m$ of the determinant of such a matrix is equal to the determinant of the matrix reduced modulo $m$ (the latter determinant being computed using modular arithmetic). In the language of category theory, the determinant is a natural transformation between the two functors $\backslash operatorname\{GL\}\_n$ and $(-)^\backslash times$.{{harvnb|Mac Lane|1998|loc=§I.4}}. See also {{section link|Natural transformation#Determinant}}. Adding yet another layer of abstraction, this is captured by saying that the determinant is a morphism of algebraic groups, from the general linear group to the multiplicative group,

:$\backslash det:\; \backslash operatorname\{GL\}\_n\; \backslash to\; \backslash mathbb\; G\_m.$

{{See also|Exterior algebra#Linear algebra}}

The determinant of a linear transformation $T\; :\; V\; \backslash to\; V$ of an $n$-dimensional vector space $V$ or, more generally a free module of (finite) rank $n$ over a commutative ring $R$ can be formulated in a coordinate-free manner by considering the $n$-th exterior power $\backslash bigwedge^n\; V$ of $V$.{{harvnb|Bourbaki|1998|loc=§III.8}} The map $T$ induces a linear map

:$\backslash begin\{align\}$

\bigwedge^n T: \bigwedge^n V &\rightarrow \bigwedge^n V \\

v_1 \wedge v_2 \wedge \dots \wedge v_n &\mapsto T v_1 \wedge T v_2 \wedge \dots \wedge T v_n.

\end{align}

As $\backslash bigwedge^n\; V$ is one-dimensional, the map $\backslash bigwedge^n\; T$ is given by multiplying with some scalar, i.e., an element in $R$. Some authors such as {{harv|Bourbaki|1998}} use this fact to define the determinant to be the element in $R$ satisfying the following identity (for all $v\_i\; \backslash in\; V$):

:$\backslash left(\backslash bigwedge^n\; T\backslash right)\backslash left(v\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_n\backslash right)\; =\; \backslash det(T)\; \backslash cdot\; v\_1\; \backslash wedge\; \backslash dots\; \backslash wedge\; v\_n.$

This definition agrees with the more concrete coordinate-dependent definition. This can be shown using the uniqueness of a multilinear alternating form on $n$-tuples of vectors in $R^n$.

For this reason, the highest non-zero exterior power $\backslash bigwedge^n\; V$ (as opposed to the determinant associated to an endomorphism) is sometimes also called the determinant of $V$ and similarly for more involved objects such as vector bundles or chain complexes of vector spaces. Minors of a matrix can also be cast in this setting, by considering lower alternating forms $\backslash bigwedge^k\; V$ with .{{harvnb|Lombardi|Quitté|2015|loc=§5.2}}, {{harvnb|Bourbaki|1998|loc=§III.5}}

The conventional definition of the determinant, as a sum over permutations over a product of matrix elements, can be written using the somewhat surprising notation of the Berezin integral. In this notation, the determinant can be written as

:$\backslash int\; \backslash exp\backslash left[-\backslash theta^TA\backslash eta\backslash right]\; \backslash ,d\backslash theta\backslash ,d\backslash eta\; =\; \backslash det\; A$

This holds for any $n\backslash times\; n$-dimensional matrix $A.$ The symbols $\backslash theta,\backslash eta$ are two $n$-dimensional vectors of anti-commuting Grassmann numbers (aka "supernumbers"), taken from the Grassmann algebra. The $\backslash exp$ here is the exponential function. The integral sign is meant to be understood as the Berezin integral. Despite the use of the integral symbol, this expression is in fact an entirely finite sum.

This unusual-looking expression can be understood as a notational trick that rewrites the conventional expression for the determinant

:$\backslash det\; A\; =\; \backslash sum\_\{\backslash sigma\; \backslash in\; S\_n\}\backslash sgn(\backslash sigma)a\_\{1,\backslash sigma(1)\}\backslash cdots\; a\_\{n,\backslash sigma(n)\}.$

by using some novel notation. The anti-commuting property of the Grassmann numbers captures the sign (signature) of the permutation, while the integral combined with the $\backslash exp$ ensures that all permutations are explored. That is, the Taylor's series for $\backslash exp$ terminates after exactly $n$ terms, because the square of a Grassmann number is zero, and there are exactly $n$ distinct Grassmann variables. Meanwhile, the integral is defined to vanish, if the corresponding Grassmann number does not appear in the integrand. Thus, the integral selects out only those terms in the $\backslash exp$ series that have exactly $n$ distinct variables; all lower-order terms vanish. Thus, the somewhat magical combination of the integral sign, the use of anti-commuting variables, and the Taylor's series for $\backslash exp$ just encodes a finite sum, identical to the conventional summation.

This form is popular in physics, where it is often used as a stand-in for the Jacobian determinant. The appeal is that, notationally, the integral takes the form of a path integral, such as in the path integral formulation for quantized Hamiltonian mechanics. An example can be found in the theory of Fadeev–Popov ghosts; although this theory may seem rather abstruse, it's best to keep in mind that the use of the ghost fields is little more than a notational trick to express a Jacobian determinant.

The Pfaffian $\backslash mathrm\{Pf\}\backslash ,A$ of a skew-symmetric matrix $A$ is the square-root of the determinant: that is, $\backslash left(\backslash mathrm\{Pf\}\backslash ,A\backslash right)^2=\backslash det\; A.$ The Berezin integral form for the Pfaffian is even more suggestive; it is

:$\backslash int\; \backslash exp\backslash left[-\; \backslash tfrac\{1\}\{2\}\; \backslash theta^T\; A\; \backslash theta\backslash right]\; \backslash ,d\backslash theta\; =\; \backslash mathrm\{Pf\}\backslash ,\; A$

The integrand has exactly the same formal structure as a normal Gaussian distribution, albeit with Grassman numbers, instead of real numbers. This formal resemblance accounts for the occasional appearance of supernumbers in the theory of stochastic dynamics and stochastic differential equations.

Determinants as treated above admit several variants: the permanent of a matrix is defined as the determinant, except that the factors $\backslash sgn(\backslash sigma)$ occurring in Leibniz's rule are omitted. The immanant generalizes both by introducing a character of the symmetric group $S\_n$ in Leibniz's rule.

For any associative algebra $A$ that is finite-dimensional as a vector space over a field $F$, there is a determinant map

{{harvnb|Garibaldi|2004}}

:$\backslash det\; :\; A\; \backslash to\; F.$

This definition proceeds by establishing the characteristic polynomial independently of the determinant, and defining the determinant as the lowest order term of this polynomial. This general definition recovers the determinant for the matrix algebra $A\; =\; \backslash operatorname\{Mat\}\_\{n\; \backslash times\; n\}(F)$, but also includes several further cases including the determinant of a quaternion,

:$\backslash det\; (a\; +\; ib+jc+kd)\; =\; a^2\; +\; b^2\; +\; c^2\; +\; d^2$,

the norm $N\_\{L/F\}\; :\; L\; \backslash to\; F$ of a field extension, as well as the Pfaffian of a skew-symmetric matrix and the reduced norm of a central simple algebra, also arise as special cases of this construction.

For matrices with an infinite number of rows and columns, the above definitions of the determinant do not carry over directly. For example, in the Leibniz formula, an infinite sum (all of whose terms are infinite products) would have to be calculated. Functional analysis provides different extensions of the determinant for such infinite-dimensional situations, which however only work for particular kinds of operators.

The Fredholm determinant defines the determinant for operators known as trace class operators by an appropriate generalization of the formula

:$\backslash det(I+A)\; =\; \backslash exp(\backslash operatorname\{tr\}(\backslash log(I+A))).$

Another infinite-dimensional notion of determinant is the functional determinant.

For operators in a finite factor, one may define a positive real-valued determinant called the Fuglede−Kadison determinant using the canonical trace. In fact, corresponding to every tracial state on a von Neumann algebra there is a notion of Fuglede−Kadison determinant.

For matrices over non-commutative rings, multilinearity and alternating properties are incompatible for {{math|n ≥ 2}},In a non-commutative setting left-linearity (compatibility with left-multiplication by scalars) should be distinguished from right-linearity. Assuming linearity in the columns is taken to be left-linearity, one would have, for non-commuting scalars a, b:

$$\backslash begin\{align\}$$

ab

&= ab \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}

= a \begin{vmatrix}1&0 \\ 0&b\end{vmatrix} \\[5mu]

&= \begin{vmatrix}a&0 \\ 0&b\end{vmatrix}

= b \begin{vmatrix}a&0 \\ 0&1\end{vmatrix}

= ba \begin{vmatrix}1&0 \\ 0&1\end{vmatrix}

= ba,

\end{align}

a contradiction. There is no useful notion of multi-linear functions over a non-commutative ring. so there is no good definition of the determinant in this setting.

For square matrices with entries in a non-commutative ring, there are various difficulties in defining determinants analogously to that for commutative rings. A meaning can be given to the Leibniz formula provided that the order for the product is specified, and similarly for other definitions of the determinant, but non-commutativity then leads to the loss of many fundamental properties of the determinant, such as the multiplicative property or that the determinant is unchanged under transposition of the matrix. Over non-commutative rings, there is no reasonable notion of a multilinear form (existence of a nonzero {{clarify span|text=bilinear form|explain=What exactly is meant by this term must be specified. This statement is valid only if the bilinear form is required to be linear on the same side for both arguments; in contrast, Bourbaki defines a bilinear form B as having the property B(ax,yb) = aB(x,y)b, i.e., left-linear in the left argument and right-linear in the other.|date=October 2017}} with a regular element of R as value on some pair of arguments implies that R is commutative). Nevertheless, various notions of non-commutative determinant have been formulated that preserve some of the properties of determinants, notably quasideterminants and the Dieudonné determinant. For some classes of matrices with non-commutative elements, one can define the determinant and prove linear algebra theorems that are very similar to their commutative analogs. Examples include the q-determinant on quantum groups, the Capelli determinant on Capelli matrices, and the Berezinian on supermatrices (i.e., matrices whose entries are elements of $\backslash mathbb\; Z\_2$-graded rings).{{Citation | url = https://books.google.com/books?id=sZ1-G4hQgIIC&q=Berezinian&pg=PA116 | title = Supersymmetry for mathematicians: An introduction | isbn = 978-0-8218-3574-6 | last1 = Varadarajan | first1 = V. S | year = 2004 | publisher = American Mathematical Soc. | postscript = .}} Manin matrices form the class closest to matrices with commutative elements.

Determinants are mainly used as a theoretical tool. They are rarely calculated explicitly in numerical linear algebra, where for applications such as checking invertibility and finding eigenvalues the determinant has largely been supplanted by other techniques."... we mention that the determinant, though a convenient notion theoretically, rarely finds a useful role in numerical algorithms.", see {{harvnb|Trefethen|Bau III|1997|loc=Lecture 1}}. Computational geometry, however, does frequently use calculations related to determinants.{{harvnb|Fisikopoulos|Peñaranda|2016|loc=§1.1, §4.3}}

While the determinant can be computed directly using the Leibniz rule this approach is extremely inefficient for large matrices, since that formula requires calculating $n!$ ($n$ factorial) products for an $n\; \backslash times\; n$ matrix. Thus, the number of required operations grows very quickly: it is of order $n!$. The Laplace expansion is similarly inefficient. Therefore, more involved techniques have been developed for calculating determinants.

Gaussian elimination consists of left multiplying a matrix by elementary matrices for getting a matrix in a row echelon form. One can restrict the computation to elementary matrices of determinant {{math|1}}. In this case, the determinant of the resulting row echelon form equals the determinant of the initial matrix. As a row echelon form is a triangular matrix, its determinant is the product of the entries of its diagonal.

So, the determinant can be computed for almost free from the result of a Gaussian elimination.

Some methods compute $\backslash det(A)$ by writing the matrix as a product of matrices whose determinants can be more easily computed. Such techniques are referred to as decomposition methods. Examples include the LU decomposition, the QR decomposition or the Cholesky decomposition (for positive definite matrices). These methods are of order $\backslash operatorname\; O(n^3)$, which is a significant improvement over $\backslash operatorname\; O\; (n!)$.{{cite arXiv|last=Camarero|first=Cristóbal|date=2018-12-05|title=Simple, Fast and Practicable Algorithms for Cholesky, LU and QR Decomposition Using Fast Rectangular Matrix Multiplication|class=cs.NA|eprint=1812.02056}}

For example, LU decomposition expresses $A$ as a product

:$A\; =\; PLU.$

of a permutation matrix $P$ (which has exactly a single $1$ in each column, and otherwise zeros), a lower triangular matrix $L$ and an upper triangular matrix $U$.

The determinants of the two triangular matrices $L$ and $U$ can be quickly calculated, since they are the products of the respective diagonal entries. The determinant of $P$ is just the sign $\backslash varepsilon$ of the corresponding permutation (which is $+1$ for an even number of permutations and is $-1$ for an odd number of permutations). Once such a LU decomposition is known for $A$, its determinant is readily computed as

:$\backslash det(A)\; =\; \backslash varepsilon\; \backslash det(L)\backslash cdot\backslash det(U).$

The order $\backslash operatorname\; O(n^3)$ reached by decomposition methods has been improved by different methods. If two matrices of order $n$ can be multiplied in time $M(n)$, where $M(n)\; \backslash ge\; n^a$ for some $a>2$, then there is an algorithm computing the determinant in time $O(M(n))$.{{harvnb|Bunch|Hopcroft|1974}} This means, for example, that an $\backslash operatorname\; O(n^\{2.376\})$ algorithm for computing the determinant exists based on the Coppersmith–Winograd algorithm. This exponent has been further lowered, as of 2016, to 2.373.{{harvnb|Fisikopoulos|Peñaranda|2016|loc=§1.1}}

In addition to the complexity of the algorithm, further criteria can be used to compare algorithms.

Especially for applications concerning matrices over rings, algorithms that compute the determinant without any divisions exist. (By contrast, Gauss elimination requires divisions.) One such algorithm, having complexity $\backslash operatorname\; O(n^4)$ is based on the following idea: one replaces permutations (as in the Leibniz rule) by so-called closed ordered walks, in which several items can be repeated. The resulting sum has more terms than in the Leibniz rule, but in the process several of these products can be reused, making it more efficient than naively computing with the Leibniz rule.{{harvnb|Rote|2001}} Algorithms can also be assessed according to their bit complexity, i.e., how many bits of accuracy are needed to store intermediate values occurring in the computation. For example, the Gaussian elimination (or LU decomposition) method is of order $\backslash operatorname\; O(n^3)$, but the bit length of intermediate values can become exponentially long.{{Cite conference

| first1 = Xin Gui

| last1 = Fang

| first2 = George

| last2 = Havas

| title = On the worst-case complexity of integer Gaussian elimination

| book-title = Proceedings of the 1997 international symposium on Symbolic and algebraic computation

| conference = ISSAC '97

| pages = 28–31

| publisher = ACM

| year = 1997

| location = Kihei, Maui, Hawaii, United States

| url = http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/ft_gateway.cfm.pdf

| doi = 10.1145/258726.258740

| isbn = 0-89791-875-4

| access-date = 2011-01-22

| archive-url = https://web.archive.org/web/20110807042828/http://perso.ens-lyon.fr/gilles.villard/BIBLIOGRAPHIE/PDF/ft_gateway.cfm.pdf

| archive-date = 2011-08-07

| url-status = dead

}} By comparison, the Bareiss Algorithm, is an exact-division method (so it does use division, but only in cases where these divisions can be performed without remainder) is of the same order, but the bit complexity is roughly the bit size of the original entries in the matrix times $n$.{{harvnb|Fisikopoulos|Peñaranda|2016|loc=§1.1}}, {{harvnb|Bareiss|1968}}

If the determinant of A and the inverse of A have already been computed, the matrix determinant lemma allows rapid calculation of the determinant of {{math|A + uv^T}}, where u and v are column vectors.

Charles Dodgson (i.e. Lewis Carroll of Alice's Adventures in Wonderland fame) invented a method for computing determinants called Dodgson condensation. Unfortunately this interesting method does not always work in its original form.{{Cite journal |last=Abeles |first=Francine F. |date=2008 |title=Dodgson condensation: The historical and mathematical development of an experimental method |url=https://www.academia.edu/10352246 |journal=Linear Algebra and Its Applications |language=en |volume=429 |issue=2–3 |pages=429–438 |doi=10.1016/j.laa.2007.11.022|doi-access=free }}

{{portal|Mathematics}}

{{colbegin}}

• Cauchy determinant
• Cayley–Menger determinant
• Dieudonné determinant
• Slater determinant
• Determinantal conjecture

{{colend}}

{{Reflist|group=nb}}

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{{Wikibooks|1= Linear Algebra

|2= Determinants

}}

{{EB1911 poster|Determinant}}

• {{SpringerEOM|title=Determinant|id=Determinant&oldid=12692|last=Suprunenko|first=D.A.}}
• {{MathWorld|title=Determinant|urlname=Determinant}}
• {{MacTutor|class=HistTopics||id=Matrices_and_determinants|title=Matrices and determinants}}
• [http://people.revoledu.com/kardi/tutorial/LinearAlgebra/MatrixDeterminant.html Determinant Interactive Program and Tutorial]
• [http://www.umat.feec.vutbr.cz/~novakm/determinanty/en/ Linear algebra: determinants.] {{Webarchive|url=https://web.archive.org/web/20081204081902/http://www.umat.feec.vutbr.cz/~novakm/determinanty/en/ |date=2008-12-04 }} Compute determinants of matrices up to order 6 using Laplace expansion you choose.
• [https://physandmathsolutions.com/Menus/matrix_determinant_calculator.php Determinant Calculator] Calculator for matrix determinants, up to the 8th order.
• [http://www.economics.soton.ac.uk/staff/aldrich/matrices.htm Matrices and Linear Algebra on the Earliest Uses Pages]
• [http://algebra.math.ust.hk/course/content.shtml Determinants explained in an easy fashion in the 4th chapter as a part of a Linear Algebra course.]

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